Topological solitons formed from scalar fields can interact with fermion fields
via the Yukawa interaction.
This fermion-soliton interaction may have a significant impact on both the
scalar field of the soliton and the fermion field.
In particular, it was shown in
Refs. [1]}, [2]}, [3]}
that the Yukawa interaction between the fermion and scalar fields of the
\((3 + 1)\) -dimensional chiral-invariant linear \(\sigma \) -model results in the
existence of the chiral soliton, a stable configuration of the interacting
scalar and fermion fields that possesses bound fermion states.
It was also shown in Refs. [4]}, [5]}, [6]} that an
external chiral field can polarize the fermion vacuum; this polarization can
be interpreted as a contribution to the baryon charge of a chiral soliton
[2]}, [3]}.
Fermion bound states also exist in the background field of a Skyrmion, and the
properties of these bound states were investigated in
Refs. [9]}, [10]}.
The main feature of the fermion-soliton interaction found in
Refs. [1]}, [2]}, [3]}, [9]}, [10]} is
that the spectrum of the Dirac operator shows a spectral flow of the
eigenvalues, where the number of zero-crossing normalized bound modes is equal
to the winding number of the soliton.
The planar baby Skyrmions inherit this feature of the fermion-soliton
interaction. It was shown in Refs. [16]}, [17]}, [18]} that in the
background field of a baby Skyrmion, the Dirac operator also shows a spectral
flow of the eigenvalues, and the number of zero-crossing normalized bounded
modes turns out to be equal to the winding number of the baby Skyrmion.
This property of the Dirac operator is preserved when the backreaction of the
fermion field coupled with the baby Skyrmion is taken into account
[19]}.
